For your tile, most full-bodied porcelains are good for outdoors, but you should ask your tile retailer if the one that you have chosen is rated for outdoor use just to be sure. The same for stone. Some are frost rated and some aren't so ask.
I agree with Rob Z about the Laticrete. I also like Laticrete products. TEC has some good stuff, but it can be pricy.
To rip up the existing tile you will need to rent a chipping gun, like a small jack hammer. If the tile is down as good as you think it is, the surface that you are left with will be rough - mostly from remaining thin-set. For that, you can either work away with your chipper or grind it off. It will be dusty so wear a mask and be thankful that you're outside.
Your other option is to leave the rough surface and go over it with a scratch coat or a self leveller before you lay the tile. It will cost you a little more for material, but save you a lot of work. If your patio slopes, you may want to stay away from the self leveller, because it will all run to the lowest corner. If it's flat this is the easiest way. If you go for the scratch coat I would use the cheapest unmodified thin-set you can find, mix it up a little on the light side and smooth it out a best as you can with a flat trowel. You'll probably want a heavy notch when you lay the tile.
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J&D Tile